A Semialgebraic Description of the General Markov Model on Phylogenetic Trees

Abstract

Many of the stochastic models used in inference of phylogenetic trees from biological sequence data have polynomial parameterization maps. The image of such a map---the collection of joint distributions for all parameter choices---forms the model space. Since the parameterization is polynomial, the Zariski closure of the model space is an algebraic variety which is typically much larger than the model space but amenable to study with algebraic methods. Of practical interest, however, is not the full variety but the subset formed by the model space. Here we develop complete semialgebraic descriptions of the model space arising from the $k$-state general Markov model on a tree, with slightly restricted parameters. Our approach depends upon both recently formulated analogues of Cayley's hyperdeterminant and on the construction of certain quadratic forms from the joint distribution whose positive (semi)definiteness encodes information about parameter values.